CONVERSE (Angles → Lines)
Goal: PROVE that Lines are Parallel (\(m \parallel n\))
Remember: The converse swaps hypothesis and conclusion to prove parallelism! Slide 5 of 12
Packet Page 2 • Guided Practice
Lines i & k cut by j
j i k A B D C E F H G
1. ∠A and ∠E: Corresponding Angles
2. ∠C and ∠E: Alternate Interior Angles
3. ∠A and ∠G: Alternate Exterior Angles
4. ∠D and ∠B: Vertical Angles
5. ∠C and ∠H: Consecutive Interior Angles
Notice: ∠D and ∠B share the same vertex → Vertical angles (always congruent!). Slide 6 of 12
Packet Page 2 • Part B
Rule: Only 2 Values Exist!
1 4 2 3 5 8 65° 7
Acute = 65° | Obtuse = \(180^\circ - 65^\circ\) = 115°
m∠1 (Obtuse): 115°
m∠2 (Acute): 65°
m∠3 (Obtuse): 115°
m∠4 (Acute): 65°
m∠5 (Obtuse): 115°
m∠8 (Acute): 65°
m∠7 (Obtuse): 115°
Every angle is either 65° or 115°. Adjacent angles always add to 180° (Linear Pair)! Slide 7 of 12
Packet Page 3 • Section 7
Step-by-Step Logic
107° 45° b° g° c° d° a° f° e°
• Angle b = 73° Linear pair with 107° (\(180 - 107\))
• Angle c = 73° Alternate Interior to angle b
• Angle d = 62° Triangle Sum (\(180 - 73 - 45\))
• Angle a = 45° Alternate Interior to 45°
• Angle e = 73° Vertical to angle c
• Angle f = 107° Alternate Exterior to 107°
• Angle g = 73° Vertical to angle b
Every angle flows logically from linear pairs, alternate interior theorems, and triangle angle sums! Slide 8 of 12
Packet Page 3 • Section 8
Corresponding Angles
2 5x + 55° 2x + 100°
Step 1: Identify Relationship
Corresponding Angles → Equal (\(\cong\))
5x + 55 = 2x + 100
3x + 55 = 100 (subtract 2x)
3x = 45 (subtract 55)
x = 15
Angle measure:
\(5(15) + 55 = \mathbf{130^\circ}\)
Linear Pair with ∠2:
m∠2 = 50°
Always plug \(x\) back in to find actual angle measures and verify linear pairs add to 180°! Slide 9 of 12
Packet Page 3 • Section 8
Two-Variable System
17x − 70 3y + 5 5y + 15 2x + 5
Solve for \(x\)
Alternate Exterior (≅)
17x − 70 = 2x + 5
15x = 75
x = 5
Angle: \(2(5) + 5 = 15^\circ\)
Solve for \(y\)
Consecutive Int. (180°)
(3y + 5) + (5y + 15) = 180
8y + 20 = 180
8y = 160
y = 20
Angles: \(65^\circ + 115^\circ = 180^\circ\)
Check: \(x = 5\) gives 15° for both exterior angles. \(y = 20\) gives 65° and 115° which sum to 180°! Slide 10 of 12
Packet Page 4 • Section 9
Which Lines Are Parallel?
n m a b 1 2 4 3 5 6 8 7 9 10 12 11 13 14 16 15
• If ∠1 ≅ ∠9: line a ∥ line b (Corr. ∠)
• If ∠9 ≅ ∠15: line n ∥ line m (Alt. Ext. ∠)
• If ∠9 ≅ ∠3: line a ∥ line b (Alt. Int. ∠)
• If ∠1 ≅ ∠11: line a ∥ line b (Corr. → Vert)
• If ∠9 & ∠16 are supp.: line n ∥ line m (Consec. Ext)
• If ∠6 & ∠13 are supp.: line a ∥ line b (Consec. Int)
Rule: Trace the two angles! The line they both touch is the transversal; the other two are the parallel lines! Slide 11 of 12
Unit Summary • Section 10
Big Takeaways
| Angle Relationship Type | Congruent or Supplementary? | Visual Memory Letter Clue |
|---|---|---|
| Corresponding Angles | CONGRUENT (\(\cong\)) | “F-Shape” — same matching corner |
| Alternate Interior Angles | CONGRUENT (\(\cong\)) | “Z-Shape” — opposite inside corners |
| Alternate Exterior Angles | CONGRUENT (\(\cong\)) | Opposite sides of transversal, outside tracks |
| Consecutive (Same-Side) Interior | SUPPLEMENTARY (180°) | “C-Shape” — interior on the same side |
| Linear Pair (Adjacent on Line) | SUPPLEMENTARY (180°) | Straight line adds to 180° |
| Vertical Angles | CONGRUENT (\(\cong\)) | “X-Shape” — opposite across vertex |
Unit complete! Keep your guided notes packet handy for reference on homework and quizzes. Slide 12 of 12
CONVERSE (Angles → Lines)
Goal: PROVE Lines are Parallel (\(m \parallel n\))
Write "congruent" / "supplementary" on the left, and "parallel" on the right. Slide 5 of 12
Packet Page 2 • Guided Practice
Interactive Prompt
j i k A B D C E F H G
1. ∠A and ∠E:
2. ∠C and ∠E:
3. ∠A and ∠G:
4. ∠D and ∠B:
5. ∠C and ∠H:
Write student responses on the board before confirming. Slide 6 of 12
Packet Page 2 • Part B
Fill-In Angle Values
1 4 2 3 5 8 65° 7
Acute = 65° | Obtuse = \(180^\circ - 65^\circ\) =
m∠1: °
m∠2: °
m∠3: °
m∠4: °
m∠5: °
m∠8: °
m∠7: °
Remind students: All 8 angles can be solved from just ONE given measure! Slide 7 of 12
Packet Page 3 • Section 7
Step-by-Step Logic
107° 45° b° g° c° d° a° f° e°
b = ° Reason: ____________________
c = ° Reason: ____________________
d = ° Reason: ____________________
a = ° Reason: ____________________
e = ° Reason: ____________________
f = ° Reason: ____________________
g = ° Reason: ____________________
Step 1: find b from linear pair; Step 2: find c from alt. interior; Step 3: find d from triangle sum! Slide 8 of 12
Packet Page 3 • Section 8
Live Work Area
2 5x + 55° 2x + 100°
Step 1: Relationship
Equation Setup & Work Area:
x = m∠2 = °
Corresponding angles are equal → \(5x + 55 = 2x + 100\). Solve for \(x\), then find \(m\angle 2\). Slide 9 of 12
Packet Page 3 • Section 8
Two Variables
17x − 70 3y + 5 5y + 15 2x + 5
Solve for \(x\)
Alt. Exterior: _________________
x =
Solve for \(y\)
Consec. Int: _________________
y =
x equations are set EQUAL; y equations are set to ADD TO 180°! Slide 10 of 12
Packet Page 4 • Section 9
Converse Reasoning
n m a b 1 2 4 3 5 6 8 7 9 10 12 11 13 14 16 15
• If ∠1 ≅ ∠9: line ∥ line
• If ∠9 ≅ ∠15: line ∥ line
• If ∠9 ≅ ∠3: line ∥ line
• If ∠1 ≅ ∠11: line ∥ line
• If ∠9 & ∠16 are supp.: line ∥ line
• If ∠6 & ∠13 are supp.: line ∥ line
Write line a, b, m, or n into the blank slots based on which transversal is shared! Slide 11 of 12
Unit Summary • Section 10
Class Wrap-Up
| Angle Relationship Type | Congruent or Supplementary? | Visual Memory Letter Clue |
|---|---|---|
| Corresponding Angles | “F-Shape” — same matching corner | |
| Alternate Interior Angles | “Z-Shape” — opposite inside corners | |
| Alternate Exterior Angles | Opposite sides of transversal, outside tracks | |
| Consecutive (Same-Side) Interior | “C-Shape” — interior on the same side | |
| Linear Pair (Adjacent on Line) | Straight line adds to 180° | |
| Vertical Angles | “X-Shape” — opposite across vertex |
Unit complete! All answers verified and ready for homework & quizzes. Slide 12 of 12
